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Consider the following two formulas for $\zeta(s)$ $$\zeta(s)=\underset{K\to\infty}{\text{lim}}\left(\frac{1}{1-2^{1-s}}\sum\limits_{n=0}^K \frac{1}{2^{n+1}}\sum\limits_{k=0}^n \binom{n}{k} \frac{(-1)^...

I have asked this question at math stack exchange, however it did not get any traction. Still curious about the answer though. Numerical evidence suggests that: $$\lim_{N \to +\infty} \sum_{n=1}^N\...